= Solution
{c}
Use <stress-energy conservation> and symmetry of the <stress-energy tensor>. The <Killing equation> for the <Killing vector field> $V$ then gives the <stress-energy current from a Killing vector>:
$$
\nabla^aJ_a=(\nabla^aT_{ab})V^b+T_{ab}\nabla^aV^b
=\frac12T^{ab}(\nabla_aV_b+\nabla_bV_a)=0.
$$
Hence $\boxed{\nabla^aJ_a=0}$. The assumption of <stress-energy conservation>, supplied by the equations of motion in part (a), is essential; the <Killing equation> alone only eliminates the second term.
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